GATE 2022 | Statistics | Question 47

Question

Let 𝑴 be any 3 × 3 symmetric matrix with eigenvalues 1, 2 and 3. Let 𝑵 be any 3 × 3 matrix with real eigenvalues such that $𝑴𝑵 + 𝑵^T𝑴 = 3𝑰$, where 𝑰 is the 3 × 3 identity matrix. Then which of the following cannot be eigenvalue(s) of the matrix 𝑵 ?

(A) 1/4 (B) 3/4 (C) 1/2 (D) 7/4

Answer 1

Since M can be any symmetric matrix with eigenvalues as 1,2,3. And N can be any matrix with real eigen values. So N can also be a symmetric matrix as real symmetric matrices have real eigenvalues.

Let $p,q,r$ be the eigenvalues of N. So we have,

$M=\begin{bmatrix} 1 & 0 &0 \\ 0&2 & 0\\ 0& 0 &3 \end{bmatrix}, N=\begin{bmatrix} p & 0 &0 \\ 0&q & 0\\ 0& 0 &r \end{bmatrix}$

$MN+N^TM=3I$

$\begin{bmatrix} 1 & 0 &0 \\ 0&2 & 0\\ 0& 0 &3 \end{bmatrix}\begin{bmatrix} p & 0 &0 \\ 0&q & 0\\ 0& 0 &r \end{bmatrix}+\begin{bmatrix} p & 0 &0 \\ 0&q & 0\\ 0& 0 &r \end{bmatrix}\begin{bmatrix} 1 & 0 &0 \\ 0&2 & 0\\ 0& 0 &3 \end{bmatrix} = 3\begin{bmatrix} 1 & 0 &0 \\ 0&1 & 0\\ 0& 0 &1 \end{bmatrix}$

$\begin{bmatrix} 2p & 0 &0 \\ 0&4q & 0\\ 0& 0 &6r \end{bmatrix} = \begin{bmatrix} 3 & 0 &0 \\ 0&3 & 0\\ 0& 0 &3 \end{bmatrix}$

$p = 3/2, q=3/4, r=1/2$

So, options A and D are correct choices

Comments

but here Matrix N can be any other matrix too, why we should take it as only symmetric and Diagonal Matrix?

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The question said that N has real eigenvalues and it is given that M is a symmetric matrix. So intuitively I choose N as symmetric as it satisfies the condition and made it diagonal to make calculations easier. If you ask me why specifically I choose this N only then there is no answer for that as intuition can only be developed with enough practice.

Answer 2

Multiply M inverse on both sides to the given equation then you will get N + N(transpose) = 3M(inverse)

not the eigenvalues of M(inverse) are $\frac{1}{1}$ , $\frac{1}{2}$, $\frac{1}{3}$

and the eigenvalues of N(transpose) are same as the eigenvalues of N thus the eigenvalues of N + N(transpose) would be twice the eigenvalues of N

now the eigenvalues of 3M(inverse) are $\frac{3}{1}$ , $\frac{3}{2}$, $\frac{3}{3}$

and we must divide that by 2 to get the eigenvalues of N that is

$\frac{3}{2}$ , $\frac{3}{4}$, $\frac{1}{2}$

A and D options are the answer